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Composition operators on weighted Bergman spaces of the polydisc

Frédéric Bayart, Anne Dorval

TL;DR

This work establishes a boundary-data–driven criterion for the boundedness of composition operators between weighted Bergman spaces on the polydisc, showing that the key condition depends only on the symbol’s behavior on the polytorus when the symbol is regular up to the boundary. It proves a stability principle across weights and provides a complete tridisc analysis, revealing that even second-order boundary data do not always suffice to decide boundedness. The authors also demonstrate that knowledge of $N$-th order derivatives at boundary points can fail to determine continuity, by constructing pairs of symbols with identical jets up to order $N$ but different mapping properties. They further characterize weight-shift possibilities on the bidisc, giving a precise description of the set of attainable target weights and identifying extreme cases, with explicit examples illustrating sharp thresholds. Overall, the paper advances the understanding of composition operators in several complex variables by linking boundary geometry to operator continuity and by detailing how weight and dimension interact in this context.

Abstract

We study composition operators between weighted Bergman spaces of the polydisc induced by smooth symbols. We prove a general result of continuity which only involves the behaviour of the symbol on the polytorus. We deduce from this several consequences about the automatic continuity of the induced operator. We study in depth the case of the tridisc and exhibit several examples showing that a characterization of continuity using only derivatives seems impossible.

Composition operators on weighted Bergman spaces of the polydisc

TL;DR

This work establishes a boundary-data–driven criterion for the boundedness of composition operators between weighted Bergman spaces on the polydisc, showing that the key condition depends only on the symbol’s behavior on the polytorus when the symbol is regular up to the boundary. It proves a stability principle across weights and provides a complete tridisc analysis, revealing that even second-order boundary data do not always suffice to decide boundedness. The authors also demonstrate that knowledge of -th order derivatives at boundary points can fail to determine continuity, by constructing pairs of symbols with identical jets up to order but different mapping properties. They further characterize weight-shift possibilities on the bidisc, giving a precise description of the set of attainable target weights and identifying extreme cases, with explicit examples illustrating sharp thresholds. Overall, the paper advances the understanding of composition operators in several complex variables by linking boundary geometry to operator continuity and by detailing how weight and dimension interact in this context.

Abstract

We study composition operators between weighted Bergman spaces of the polydisc induced by smooth symbols. We prove a general result of continuity which only involves the behaviour of the symbol on the polytorus. We deduce from this several consequences about the automatic continuity of the induced operator. We study in depth the case of the tridisc and exhibit several examples showing that a characterization of continuity using only derivatives seems impossible.
Paper Structure (13 sections, 29 theorems, 228 equations)

This paper contains 13 sections, 29 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. Some general results
  3. Notations
  4. Technical lemmas
  5. How to prove continuity
  6. Applications
  7. N-th order derivatives are not enough
  8. Two different weights
  9. About continuity on the tridisc
  10. Examples
  11. Applications to composition operators on the bidisc
  12. On the range of composition operators
  13. A characterization of the extreme cases

Key Result

Theorem 1.1

Let $n\geq 1$. There exist $\phi,\psi\in\mathcal{O}(\mathbb D^3,\mathbb D^3)\cap\mathcal{C}^\infty(\overline{\mathbb D}^3)$ and $\beta\in(-1,0)$ such that

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark
  • Lemma 2.5
  • ...and 52 more